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.