Let A ∈ Rm×n, B ∈ Rn×r, and C = AB. Show that (a) the column space of C is a subspace of the column space of A. (b) the row space of C is a subspace of the row space of B. (c) rank(C) ≤ min{rank(A), rank(B)}.

Verify whether the subsets of W of V = M₂2 is a subspace: ² = * = ²x] = (a) Closure Property of Vector Addition W (B) Closure Property of Scalar Multiplication W is A SUBSPACE/NOT A SUBSPACE of V = M₂2.

Let P₂ be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned by 14x² + 12x +11, 9x² + 8x +7 and - (22x² + 20x +17). a. The dimension of the subspace His b. Is {14x² + 12x + 11, 9x² + 8x +7, − (22x² + 20x +17)} a basis for P₂? choose c. A basis for the subspace H is { Be sure you can explain and justify your answer. }. Enter a polynomial or a comma separated list of polynomials, where you can enter xx in place of ².

Show that the set W of vectors (x,y,z) in R3 such that x = 3y is a subspace of R3 (Note: you MUST be clear and concise in your notation and go through each step thoroughly).

Find a linearly independent set of vectors that spans the same subspace of R4 as that spanned by the vectors A linearly independent spanning set for the subspace is: -4 3 1 2 -5 4