What is the difference between matrix algebra and linear algebra

A similar remark applies in general: Matrix products can be written unambiguously with no parentheses. However, a note of caution about matrix multiplication must be taken: The fact that and need not be equal means that the order of the factors is important in a product of matrices. For example and may not be equal. Warning: If the order of the factors in a product of matrices is changed, the product matrix may change or may not be defined.

Ignoring this warning is a source of many errors by students of linear algebra! Properties 3 and 4 in Theorem 2. They assert that and hold whenever the sums and products are defined. These rules extend to more than two terms and, together with Property 5, ensure that many manipulations familiar from ordinary algebra extend to matrices. For example. Note again that the warning is in effect: For example need not equal. These rules make possible a lot of simplification of matrix expressions.

Example 2. Matrices and are said to commute if. Solution: Showing that commutes with means verifying that. The computation uses the associative law several times, as well as the given facts that and. Hence if , then follows.

Conversely, if this last equation holds, then equation 2. This gives , and follows. Thus the system of linear equations becomes a single matrix equation. Matrix multiplication can yield information about such a system. Suppose that is any solution to the system, so that. Multiply both sides of this matrix equation by to obtain, successively,.

This shows that if the system has a solution , then that solution must be , as required. But it does not guarantee that the system has a solution. However, if we write , then. Thus will be a solution if the condition is satisfied.

The ideas in Example 2. Three basic operations on matrices, addition, multiplication, and subtraction, are analogs for matrices of the same operations for numbers. In this section we introduce the matrix analog of numerical division.

To begin, consider how a numerical equation is solved when and are known numbers. If , there is no solution unless. But if , we can multiply both sides by the inverse to obtain the solution. Of course multiplying by is just dividing by , and the property of that makes this work is that. This suggests the following definition.

A matrix that has an inverse is called an. Note that only square matrices have inverses. Even though it is plausible that nonsquare matrices and could exist such that and , where is and is , we claim that this forces. Indeed, if there exists a nonzero column such that by Theorem 1.

Similarly, the condition implies that. Hence so is square. Compute and. Hence , so is indeed an inverse of. Solution: Let denote an arbitrary matrix. Hence cannot equal for any. The argument in Example 2. But Example 2. However, if a matrix does have an inverse, it has only one. Since and are both inverses of , we have. If is an invertible matrix, the unique inverse of is denoted.

Hence when it exists is a square matrix of the same size as with the property that. These equations characterize in the following sense:. Inverse Criterion: If somehow a matrix can be found such that and , then is invertible and is the inverse of ; in symbols,. This is a way to verify that the inverse of a matrix exists. We have , and so. Hence , as asserted. This can be written as , so it shows that is the inverse of.

That is,. The next example presents a useful formula for the inverse of a matrix when it exists. To state it, we define the and the of the matrix as follows:. Solution: For convenience, write and. Then as the reader can verify. So if , scalar multiplication by gives. Hence is invertible and.

Thus it remains only to show that if exists, then. We prove this by showing that assuming leads to a contradiction.

In fact, if , then , so left multiplication by gives ; that is, , so. But this implies that , , , and are all zero, so , contrary to the assumption that exists. As an illustration, if then. Hence is invertible and , as the reader is invited to verify. The determinant and adjugate will be defined in Chapter 3 for any square matrix, and the conclusions in Example 2.

Matrix inverses can be used to solve certain systems of linear equations. Recall that a of linear equations can be written as a matrix equation. If is invertible, we multiply each side of the equation on the left by to get. This gives the solution to the system of equations the reader should verify that really does satisfy. Furthermore, the argument shows that if is solution, then necessarily , so the solution is unique.

Of course the technique works only when the coefficient matrix has an inverse. This proves Theorem 2. If the coefficient matrix is invertible, the system has the unique solution. In matrix form this is where , , and. Then , so is invertible and by Example 2. Thus Theorem 2.

If a matrix is and invertible, it is desirable to have an efficient technique for finding the inverse. The following procedure will be justified in Section 2. First interchange rows 1 and 2. Next subtract times row 1 from row 2, and subtract row 1 from row 3. Hence , as is readily verified. Given any matrix , Theorem 1. If , the matrix is invertible this will be proved in the next section , so the algorithm produces. If , then has a row of zeros it is square , so no system of linear equations can have a unique solution.

But then is not invertible by Theorem 2. Hence, the algorithm is effective in the sense conveyed in Theorem 2. Let be an invertible matrix. Show that:. If , then. Given the equation , left multiply both sides by to obtain. Thus , that is.

This proves 1 and the proof of 2 is left to the reader. Properties 1 and 2 in Example 2. Here is a specific example:. Sometimes the inverse of a matrix is given by a formula. The idea is the : If a matrix can be found such that , then is invertible and. Its transpose is the candidate proposed for the inverse of. Using the inverse criterion, we test it as follows:. Hence is indeed the inverse of ; that is,. We are given a candidate for the inverse of , namely.

We test it as follows:. Hence is the inverse of ; in symbols,. If is invertible, so is , and. If and are invertible, so is , and. If are all invertible, so is their product , and. If is invertible, so is for any , and. If is invertible and is a number, then is invertible and. If is invertible, so is its transpose , and. Proof: 1. This is an immediate consequence of the fact that. The equations show that is the inverse of ; in symbols,. Use induction on.

If , there is nothing to prove, and if , the result is property 3. We apply this fact together with property 3 as follows:. This is property 4 with. The reversal of the order of the inverses in properties 3 and 4 of Theorem 2. Another manifestation of this comes when matrix equations are dealt with. If a matrix equation is given, it can be by a matrix to yield. Similarly, gives.

However, we cannot mix the two: If , it need be the case that even if is invertible, for example, ,. Part 7 of Theorem 2. Hence , so by Theorem 2. The following important theorem collects a number of conditions all equivalent to invertibility.

It will be referred to frequently below. The following conditions are equivalent for an matrix :. The homogeneous system has only the trivial solution. The system has at least one solution for every choice of column. There exists an matrix such that. If exists, then gives. Assume that 2 is true. Certainly by row operations where is a reduced, row-echelon matrix. It suffices to show that. Suppose that this is not the case. Then has a row of zeros being square. Now consider the augmented matrix of the system.

Then is the reduced form, and also has a row of zeros. Since is square there must be at least one nonleading variable, and hence at least one parameter. Hence the system has infinitely many solutions, contrary to 2.

So after all. Consider the augmented matrix of the system. Using 3 , let by a sequence of row operations. Then these same operations carry for some column. Hence the system has a solution in fact unique by gaussian elimination. This proves 4. Write where are the columns of. Now let be the matrix with these matrices as its columns. Assume that 5 is true so that for some matrix. Then implies because. Thus condition 2 holds for the matrix rather than. Hence the argument above that 2 3 4 5 with replaced by shows that a matrix exists such that.

But then. Thus which, together with , shows that is the inverse of. This proves 1. The proof of 5 1 in Theorem 2. We record this important fact for reference. Here is a quick way to remember Corollary 2. If is a square matrix, then. If then. Observe that Corollary 2. For example, we have.

In fact, it can be verified that if and , where is and is , then and and are square inverses of each other. An matrix has if and only if 3 of Theorem 2. Skip to content Introduction In the study of systems of linear equations in Chapter 1, we found it convenient to manipulate the augmented matrix of the system.

Arthur Cayley showed his mathematical talent early and graduated from Cambridge in as senior wrangler. With no employment in mathematics in view, he took legal training and worked as a lawyer while continuing to do mathematics, publishing nearly papers in fourteen years.

Finally, in , he accepted the Sadlerian professorship in Cambridge and remained there for the rest of his life, valued for his administrative and teaching skills as well as for his scholarship. His mathematical achievements were of the first rank. In addition to originating matrix theory and the theory of determinants, he did fundamental work in group theory, in higher-dimensional geometry, and in the theory of invariants. He was one of the most prolific mathematicians of all time and produced papers.

Given , and discuss the possibility that , ,. If and are matrices of the same size, their sum is the matrix formed by adding corresponding entries. If and , compute. Find , , and if. Let , ,. User Info: LordOfDabu. I have discovered a truly marvelous proof, which this signature is too narrow to contain. It's only been in the last few years that you've seen a large increase in the of people going for that undergrad specialization.

That's because there's been an increased need for those majors and the opportunities for those jobs have been increased.

That really isn't a fad, its more of a progression. More topics from this board Why did this board die? Ask A Question. Browse More Questions. Keep me logged in on this device. Forgot your username or password? User Info: Shinwa Shinwa 13 years ago 1 They seem very similar, but both the college courses Linear Algebra and Matrix Algebra both use a linear algebra textbook, so what is the exact difference? The Linear Algbra course has a prerequisite in a discrete math type course. Improve this question.

Pete L. Clark It's a matter of emphasis, really. Linear algebra cares about those, but also rational canonical forms, etc Add a comment. Active Oldest Votes. Improve this answer. Qiaochu Yuan Qiaochu Yuan k 34 34 gold badges silver badges bronze badges. Vectors are mathematical object living in a linear space or vector space which satisfy certain properties.

Choosing a special set of vectors called a base, we can decompose every vector in the vector space into a kind of sum of vectors in this base. Thus every vector in a code, and this is the row collum matrix. The next step is to look at the homomorphisms maps between linear spaces. I'd argue this is what a matrix really is and that ordering is an artifact of trying to write something in linear order on a piece of paper.

Irving Kaplansky, writing of himself and Paul Halmos. John Stillwell John Stillwell Thanks for sharing. In my opinion, 'abstract' is not automatically 'better. It's possible to do a heck of a lot of symbolic calculation in such settings through the judicious use of integral matrices here "integral" should be considered broadly. Konrad Waldorf Konrad Waldorf 3, 4 4 gold badges 25 25 silver badges 31 31 bronze badges. See, for example, Loewner's classification of matrix-monotone functions, or most any paper in quantum Shannon theory.

So instead of trying so hard to misunderstand him, try to find a meaning in his comment. It's not an arrogant statement about how easy he thinks linear algebra is, but rather a castigation of those "generations of professors and textbook writers" who turned an elegant subject into a jumbled mess.

Show 5 more comments. Steve Huntsman Steve Huntsman Jon Jon 2 2 silver badges 7 7 bronze badges.