Prove that the given set with the stated operations is a vector space. The set Mm n of all m ×n

Prove that the given set with the stated operations is a...

Key Concepts

Vector Space

A vector space is a collection of objects, called vectors, that can be added together and multiplied by scalars while satisfying a set of axioms. These axioms include closure under addition and scalar multiplication, the existence of an additive identity and inverses, associativity and commutativity of addition, and distributivity of scalar multiplication over both vector addition and field addition. This concept is central to linear algebra and underlies many structures and methods in mathematics and applied fields.

Matrices as Elements of the Space

In the context of this question, the elements of the vector space are m×n matrices. Each matrix is a rectangular array of elements (often numbers) and is treated as a vector in this setting. This approach allows techniques from linear algebra to be applied to matrices, enabling operations such as solving systems of linear equations and transformations in higher-dimensional spaces.

Matrix Addition

Matrix addition is defined by adding corresponding entries of two matrices of the same size. This operation is both commutative and associative, and it ensures that the sum of any two m×n matrices is another m×n matrix. It plays a critical role in satisfying the vector space axiom that the set is closed under addition.

Scalar Multiplication

Scalar multiplication for matrices involves multiplying every entry of a matrix by the same scalar from the underlying field. This operation satisfies important properties such as distributivity over both scalar and matrix addition, and compatibility with field multiplication, which are required by the vector space axioms.

Verification of the Vector Space Axioms

Proving that the set of all m×n matrices forms a vector space involves checking the vector space axioms. Each property, like closure, existence of an additive identity (zero matrix), existence of additive inverses, associativity, and distributivity, must be verified for the operations of matrix addition and scalar multiplication. This verification process is a key concept in understanding the structure of vector spaces and demonstrating that matrices behave as expected under these operations.

Transcript

00:01 In this case we got the set of m by n matrices and is defined with the usual summation and multiplication.

00:09 So we need to prove that this space m is a vector space.

00:14 So just to remember that the usual summation in by matrices means component by component.

00:22 Here you can observe that we got a11, b1 and what we does in the resultant matrix is, is a11 plus b11, a1n plus b1n and so on.

00:36 So basically we're summing each of the components in the respective positions of the matrix.

00:45 And the scalar product is similar.

00:49 We are operating.

00:51 We're multiplying the scalar by each of the components that appear on the matrix.

00:57 Ok, so that's the definition of the scalar multiplication and matrix summation.

01:05 However, this kind of notation is kind of difficult to manage and for that is better to determine a new notation.

01:16 So we can characterize a matrix a by its components.

01:20 That means that we choose a a j, where i goes from one to m and j goes from one to n.

01:28 So basically we're defining here all the matrix by all its components and this help us a lot to to to handle better with operations and instead of writing these big matrices that makes the things difficult so in that case the summation of matrices will be just component by component these indices should be coincided with these ones and the same for this kind of multiplication is just multiplying k by each of the components remember that from now on i will go from 1 to m and j will go from 1 to n so i just need to put ij on each of the coefficients this this a ij will be the coefficients of the matrices and it's better to put in this notation okay so now now that we have defined that you can observe actually just up here that when you sum two matrices what you obtain is again a matrix that is an n by n matrix so that means that this the sum is part of the same space of n by n matrices so it's part of our vector space that is related with the first axiom of vector vector spaces of the closure of the addition operation and the same happened here with the scalar multiplication what happened is that you obtain again a matrix that is an n by n matrix that means that this is again an m n by n so it's part of the same space this proof the sixth action of vector spaces so let's continue from the from the second one so we know we're going to use our new notation to handle better these operations here so what happened is that in the left -hand side we obtain a ij plus m b ij and on the right -hand side we obtain b -j plus a ij and what happened here is that each of the components of the matrix is a real number so the sum of two real numbers commute that means that we can exchange this to obtain a ij plus mvd and that's the same as the left hand side.

04:05 So this are some holes.

04:07 Just to visualize what's happening here is that in the matrix, in the results on the matrix on the left hand side of the expression, you obtain a11 plus b1 and so on.

04:23 And on the right hand side what you obtain here is b11 plus a11 and so on for the rest of the elements and the point is that these elements are real so they can be exchanged it so you obtain a11 plus b11 and the same for the rest of the elements and that's the same as the left -hand side therefore this axiom holds on these vector space of matrices so this just to visualize what i am doing here with the indices with the index notation.

05:04 Let's continue.

05:05 So now we got a plus b plus c.

05:09 This is related with the associativity property.

05:12 So again, here we can write as just the coefficients as in this with the index notation, plus and here we have b -i -j plus c -i -j...

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