Let A = 3 1 1 2 0 1 4 2 1 1 -1 1 Find a basis of the null space of A. Find the dimension of the null space, and find the rank of A. Enter the basis for the null space of A as a collection of one or more vectors, using "(" and ")" as enclosing brackets, separating each vector with a comma. } { What is dim [null(A)]? } What is the rank of A?

Let A = - - ст 5 -5 -5 15 -4 -6 -2 -5 1 - -3 -6 0 3 7 - -1 Find a basis for the column space of A.

3. orthonormal basis B generated by B.

Apply Gram-Schmidt to (1, 0, −1, 1), (2, 1, −1, 0) , (2, −1, −1, 3) to obtain an orthonormal set with the same span as these vectors.

Find a basis for Row(A) and for Col(A). Row(A) Col(A) A = 9 4 -5 19 8 1 4 3 118 ↓ 1 0 OF}

Say you are given two vectors x and y represented using different basis sets {ê;} and {f;}, of the same dimension N. That is, x = y = E y;f;. E xie; and N you are told that a given vector x has the representation x = (1, 1)" in basis {ê;} and the representation x = (-1,1)" in basis {f}}. How might the two basis sets be related? Draw a figure to justify X = your answer.

Let B={v_1...v_n} be some basis for R^n. Prove that if x is an element of R^n then x can be expressed as a linear combination of the basis vectors in a unique way. (i.e. if x=a_1v_1+...+a_nv_n and x=b_1v_1+...+b_nv_n then a_i=b_i for each i.)