5. True or False? (1 point each) Either the row vectors or the column vectors of

Question

5. True or False? (1 point each)

Either the row vectors or the column vectors of a square matrix are linearly independent. A matrix with linearly independent row vectors and linearly independent column vectors Is square. The dimension of the nullspace of a nonzero m times n matrix is at most m. Adding one additional column to a matrix increases its rank by one. The dimension of the nullspace of a square matrix with linearly dependent rows is at least one. If A is square and Ax = b does not have a solution for some vector b, then the dimension of the nullspace of A Is zero. If a matrix .4 has more rows than columns, then the dimension of the row space is greater than the dimension of the column space. If rank(A^T)=rank(A), then A is square. There is no 3 times 3 matrix whose row space and nullspace are both lines in 3-space. If V is a subspace of R^n and W a subspace of V, then W^, is a subspace of V^

Explanation / Answer

a) either the row vectors or the column vectors of a square matrix are linearly independent ----- True

b) a matrix with linearly independent row vectors and linearly independent column vectors is square ---- True

c) the dimension of the nullspace of nonzero m*n matrix is at most m ----- False

d) adding one additional column to a matrix increases its rank by one ---- false

e) the dimension of the nullspace of a square matrix with linearly independent rows is at least one --- true

f) if A is square matrix and Ax=b does not have a solution for some vector b, then the dimension of the null space of A is zero. ---- true

g) if A is square matrix and Ax=b does not have a solution for some vector b, then the dimension of the null space of A is zero. ---- false

h) if rank of A^T is equal to rank of A, then A is square ---false

i)there is no 3x3 matrix whose row space and nullspace are both lines in 3-space----true

j) if V is subspace of Rn and W a subspace of V,then W1 iis a subspace of V1------true

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