Exam I

Any row vector is in row-echelon form.

True.

The span of two parallel vectors in R^2 is R^2

False. You need two non-parallel vectors to span R^2.

The value of r for which the vector v = [r,4,0] is in the span of S = { [-1, 1,1],[2,-3,1]} is.

-3

What is the minimum possible nullity for a 7x4 matrix?

0. The maximum possible rank is 4 because that is the most nonzero rows you can have in a matrix with 4 rows (since the leading entries have to move to the right). That gives minimum nullity = 4 - 4 = 0

If B is a 3x4 matrix, then its rows are 4x1 vectors.

False. A row in a 3x4 matrix has 4 horizontal entries. Therefore they are 1x4 vectors, not 4x1 vectors.

If a matrix vector product Av = 0 , then v must be the zero vector.

False. Yes, not necessarily zero vector: for example A could have been a zero matrix. There are other possibilities too. Ex. A = [1,1;1,1] v = [1;-1]

The equation Ax = b is consistent if and only if b is a linear combination of columns of A.

True. If the equation Ax = b is consistent, it has a solution x = c so that Ac = b. That gives b as a linear combination of columns of A because the matrix vector product Ac is, by definition, a linear combination of columns of A.

The rank of a matrix equals the number of pivot columns of the matrix.

True. Rank is the number of non-zero rows in the row-echelon form. Each nonzero row would contain a pivot, and therefore corresponds to a pivot column.

The identity matrix is a square matrix.

True. The identity matrix consists of the n standard unit vectors of the n-dimensional space. This gives an nxn matrix.

Performing elementary row operations on the augmented matrix of a system of linear equations, produces the augmented matrix of an equivalent system of linear equations.

True. The systems remain equivalent after elementary operations.