Q1. Find the bases and dimensions for Row space, Column space and Null space of the...

 

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Q1. Find the bases and dimensions for Row space, Column space and Null space of the matrix: -2 3 7 1 -2 8 A = -3 3 8 4 4 -2 1 -5 -7] Q2. Let V = {(a, b, c, d) : b+ 2c + d = 0} and W = {(a, b,c, d) : a + 2b = 0, c = 5d} be two subspaces of R*. Find the bases and dimensions for V, W,V + W and V n W. Q3. If T: R* → R3 be a linear transformation defined by: T(x, y,z, t) = (x + 2y + 3z + 2t, 2x + 4y + 7z + 5t, x + 2y + 6z + 5t), then find the dimensions and bases of Kernel and Image of T. Q4. Consider the following two bases of R²: S = {u1, u2} = {(1,2), (3,5)} and S' = {V,, v2} = {(1,-1), (1, -2)} (a) Find the change-of-basis matrix P from S to the "new" basis S'. (b) Find the change-of-basis matrix Q from the "new" basis S' back to the "old" basis S. Q5. Let S and T be linear transformations from R3 to R3 defined as follows: S(x, y, z) = (3x + 5y + z, –2x + y – z, x – z) and T(x, y,z) = (x – y,y - z,z – x). Compute the associated matrices [S]a [T]a, [S + T]a, [SoT]a and [T-1la if T-1 exists, where a is the standard basis of R. 3 42 Q1. Find the bases and dimensions for Row space, Column space and Null space of the matrix: -2 3 7 1 -2 8 A = -3 3 8 4 4 -2 1 -5 -7] Q2. Let V = {(a, b, c, d) : b+ 2c + d = 0} and W = {(a, b,c, d) : a + 2b = 0, c = 5d} be two subspaces of R*. Find the bases and dimensions for V, W,V + W and V n W. Q3. If T: R* → R3 be a linear transformation defined by: T(x, y,z, t) = (x + 2y + 3z + 2t, 2x + 4y + 7z + 5t, x + 2y + 6z + 5t), then find the dimensions and bases of Kernel and Image of T. Q4. Consider the following two bases of R²: S = {u1, u2} = {(1,2), (3,5)} and S' = {V,, v2} = {(1,-1), (1, -2)} (a) Find the change-of-basis matrix P from S to the "new" basis S'. (b) Find the change-of-basis matrix Q from the "new" basis S' back to the "old" basis S. Q5. Let S and T be linear transformations from R3 to R3 defined as follows: S(x, y, z) = (3x + 5y + z, –2x + y – z, x – z) and T(x, y,z) = (x – y,y - z,z – x). Compute the associated matrices [S]a [T]a, [S + T]a, [SoT]a and [T-1la if T-1 exists, where a is the standard basis of R. 3 42