4.5 Flashcards
what is the dimension in a subspace?
the number of vectors in a subspace or the number of pivots when those vectors form a matrix. It’s also the n in R^n.
Ex. If the number of vectors in the calculated basis of the given subspace is 2, the dimension of the given subspace must also be 2. If theres 2 pivots in the matrix’s echelon form, then the dim is 2. If the subspaces spans R^2 then the dim of subspace = 2
T or F: Let A be an m x n matrix.
The number of variables in the equation Ax = 0 equals the nullity of A.
This statement is false. The number of free variables is equal to the nullity of A.
T or F: Let A and B be m x n matrices.
If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.
The statement is false. The columns of an echelon form B of A are often not in the column space of A.
T or F: Let A be an m x n matrix.
The nullity of A is the number of columns of A that are not pivot columns.
The statement is true. The nullity of A equals the number of free variables in the equation Ax = 0.
T or F: Let V be a nonzero finite-dimensional vector space, and the vectors listed belong to V.
If there exists a set {v_1, … , v_p} that spans V, then dim V ≤ p.
This statement is true. Apply the Spanning Set Theorem to the set {v_1, … , v_p} and produce a basis for V. This basis will not have more than p elements in it, so dim V ≤ p.
T or F: Let V be a nonzero finite-dimensional vector space, and the vectors listed belong to V.
If there exists a linearly independent set {v_1, … , v_p} in V, then dim V ≥ p.
This statement is true. The span of the set {v_1, … , v_p} will be a subspace of V which can be expanded to find a basis for V. This basis will contain at least p elements, so dim V ≥ p.
If C has m cols and n rows and m > n, then
not all cols have pivots which means linearly dependent cols which means non trivial soln
what’s the standard basis?
basis made up of the elementary vectors that span R^n
what’s rank thrm?
rank A (# of pivots) + Nul A (# of free variables) = # of cols in A
