4. [15 points) Consider the following matrices: v1 = 4 6 2 2 3 1 (6 9 3 / 2 ,02 = 1 1 | 3 -2 -1 -3 2 1 3 , V3 = 12 2 2 6 1 1 | 18 3 3 , 7 -5 3 -3 1 4 -2 2 76 -3 -2 w1 = 3 -1 3 ,w2 = -2 0 -2 , W3 = -1 -2 2 -2 2 1 3 -3 -2 2 -2 -3 1-3 3 -1 - 7 -3 2 3 -1 0 w4=| 1 -1 1), W5 = -3 -1 -2 , W6 = | -2 0 0 -3 3 -2 / -2 2 2 1 -1 3 Let V = span{V1, V2, V3} and W = span{W1, W2, W3, W4, W5, W6}. (a) Use the row space of a matrix to find simpler bases for V and W. By looking at these bases, give a simple description in words of the subspaces V and W. Hint: W can be described by 3 independent conditions. (b) Find a basis of V +W. (c) Find a basis of V W. (It might help to find the dimension first!) 4. [15 points) Consider the following matrices: v1 = 4 6 2 2 3 1 (6 9 3 / 2 ,02 = 1 1 | 3 -2 -1 -3 2 1 3 , V3 = 12 2 2 6 1 1 | 18 3 3 , 7 -5 3 -3 1 4 -2 2 76 -3 -2 w1 = 3 -1 3 ,w2 = -2 0 -2 , W3 = -1 -2 2 -2 2 1 3 -3 -2 2 -2 -3 1-3 3 -1 - 7 -3 2 3 -1 0 w4=| 1 -1 1), W5 = -3 -1 -2 , W6 = | -2 0 0 -3 3 -2 / -2 2 2 1 -1 3 Let V = span{V1, V2, V3} and W = span{W1, W2, W3, W4, W5, W6}. (a) Use the row space of a matrix to find simpler bases for V and W. By looking at these bases, give a simple description in words of the subspaces V and W. Hint: W can be described by 3 independent conditions. (b) Find a basis of V +W. (c) Find a basis of V W. (It might help to find the dimension first!)