5 True or False 1 point each Either the row vectors or the c

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^

Solution

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 W^{1 i}is a subspace of V¹------true