The subspaces of R3 are planes, lines, R3 itself, or Z containing only (0, 0, 0).(a) Describe the three types of subspaces of R2 .(b) Describe all subspaces of D, the space of 2 by 2 diagonal matrices.

Let U be the space of 3 x 3 symmetric matrices and V be the space of 3 x 3 skew symmetric matrices. Show that dim U + dim V = 3².

Let the Matrix A =( 1 -2 -6 , 0 3 7 , 1 -2 5 ) and vector b = (11 , -5 , 9 ) is b in the span of the column space A ?

2 and u = 4 Write as the sum of two orthogonal vectors, one in Span {u} and one orthogonal to u. Let y = - 8 y=ý+z=D+0 (Type an integer or simplified fraction for each matrix element. List the terms in the same order as they appear in the original list.)

The set of non-zero vectors {b,, b,, bz, b4} are vectors in R* with the property that 4b, + 2b, = 6b, . Let matrix B = | b, b2 b; b4. (a) What does the matrix b, b, b, reduce to in RREF form? (b) Find a specific solution to the matrix equation Bx = 0. That is, find an x vector that solves that equation. (c) What is the maximum number of pivots matrix B can have? Please explain briefly referencing anything from above. (d) Can the set of vectors {b1, b2, b3, b4} span R’? Explain why or why not? (e) Can the set of vectors {b,, b2, b3, b4} span R*? Explain why or why not? (f) Matrix M is 7 x 4. If possible, show that the columns of the matrix MB are linearly dependent. If it is not possible to show this, explain why.