Question

# Suppose that \mathbf{A}=\left(\begin{array}{cccc} 1 & x & y & 1 \\ 1 & x & x & x \\ x & 1 & x y & y \\ x & x & x y & 1 \end{array}\right) \in \mathscr{M}_{4,4}(\mathbb{R}) Show that A is not invertible if and only if either x = y, x =1 or xy = 1. Let V and W be 3-dimensional vector spaces over R, Bị = {V1, V2, V3} be an ordered basis for V and B2 = {w1,w2, w3} be an ordered basis for W. Define a linear transformation T :V → W by T\left(\mathbf{v}_{1}\right)=\mathbf{w}_{1}+\mathbf{w}_{2} T\left(\mathbf{v}_{2}\right)=\mathbf{w}_{1}-\mathbf{w}_{2}+\mathbf{w}_{3} T\left(\mathbf{v}_{3}\right)=\mathbf{w}_{3} (i) Write down the matrix A representing T with respect to the ordered bases B1 and (ii) Calculate the dimension of the kernel of T and of the image of T. (iii) Show that B3 = {v1,V1 + V2, V1 + V2 + V3 } is a basis for V and B4 = {w1 - W2, W2 +W3, W1 – W3} is a basis for W. ) Find the matrix which represents T with respect to the ordered bases B3 and B4.  Fig: 1  Fig: 2  Fig: 3  Fig: 4  Fig: 5  Fig: 6  Fig: 7  Fig: 8  Fig: 9  Fig: 10  Fig: 11  Fig: 12