Question

We have two linear representations

R³→R²,
T: R3 -> R2, S: R2 ->R3
where T is given by matrix A and S by matrix B:
A=[ 1 2 -1
2 1 3]
a) FInd the matrix products AB and BA
b) find x
B=[ 1 2 ]
[ -5 0 ] x=T(3/-1/-2),og y = S(x)
[ 3 1 ]
c) Let R be the composite

Answer 1

To find matrix products AB and BA, one must multiply matrix A by matrix B and vice versa, respectively; vector x is found by applying transformation T to the given vector, then y is obtained by applying S to x.

Step-by-step explanation:

The question involves two linear transformations T : R^3 \rightarrow R^2 and S : R^2 \rightarrow R^3, represented by matrices A and B respectively. The matrix products AB and BA are to be found, followed by the image of a given vector under these transformations and the resultant composite.

For matrix A, which is a 2x3 matrix, and matrix B, which is a 3x2 matrix, the matrix product AB will be a 2x2 matrix, whereas the matrix product BA will be a 3x3 matrix. This also demonstrates the composability of linear transformations.

To find the products, we perform matrix multiplication according to the standard rules where each entry in the resulting matrix is the dot product of the corresponding row of the first matrix and the column of the second.

In the case of applying the transformation T to vector (3, -1, -2) to get vector x, we use matrix A to transform this vector. This is done by multiplying the matrix A by the vector. Afterwards, the transformation S is applied, represented by matrix B, to get vector y, achieved by multiplying matrix B with vector x.

The properties of scalar and vector products discussed are not directly applicable to this question as they relate to operations in vector space, whereas the question asks for simple matrix multiplication.

Answer 2

To find the matrix products AB and BA, multiply the matrices A and B in the correct order. To find x and y, substitute the given values into the equations x = T(3, -1, -2) and y = S(x). The composite representation R is found by multiplying SB.

Step-by-step explanation:

To find the matrix products AB and BA, we need to multiply the matrices A and B in the correct order.

Multiplying AB:

1. Multiply the first row of A by the first column of B to get the first element of AB.
2. Multiply the first row of A by the second column of B to get the second element of AB.

Multiplying BA:

1. Multiply the first row of B by the first column of A to get the first element of BA.

So, AB = [1(1) + 2(-5) + (-1)3, 1(2) + 2(0) + (-1)1] = [-2, 0] and BA = [1(1) + 2(2), -5(1) + 0(2), 1(3) + (-1)(2)] = [5, -5, 1].

To find x, we substitute the given values of T into x = T(3, -1, -2).

1. Calculate T(3, -1, -2) = [1(3) + 2(-1) + (-1)(-2), 2(3) + 1(-1) + 3(-2)] = [1, -4].

To find y, we substitute the calculated value of x into y = S(x).

1. Calculate S([1, -4]) = [1(1) + 2(-4), -5(1) + 0(-4), 3(1) + 1(-4)] = [-7, -5, -1].

Finally, the composite representation R is R = SB = [-7, -5, -1]B = [-7(1) + (-5)(-5) + (-1)3, -7(2) + (-5)(0) + (-1)1, -7(3) + (-5)2 + (-1)] = [-4, -14, -26].