LINEAR ALGEBRA PRACTICE EXAM 1
QUESTIONS AND ANSWERS
The span of ⃗0 in Rn is ... - Answer-the single vector ⃗0.
The span of a single nonzero vector in R2 and R3 is ... - Answer-a line through ⃗0.
The span of two nonzero vectors ⃗u and ⃗v in R3, with ⃗v ̸= c⃗u, is ... - Answer-a plane
through ⃗0.
An indexed set of vectors S = {⃗v1,⃗v2,...,⃗vp} in Rn is said to be linearly independent (we
may say also that the vectors ⃗v1,...,⃗vp are linearly independent) if the vector equation
x1⃗v1 +x2⃗v2 +···+xp⃗vp =⃗0
has only the __________ solution - Answer-trivial
Recall that the vector equation is equivalent to the homogeneous equation A⃗x = ⃗0 and
he matrix equation A⃗x = ⃗0 has a nontrivial solution if and only if it has a ___________. -
Answer-free variable
The columns of a matrix A are linearly independent exactly when the matrix equation A⃗x
= ⃗0 has
only the trivial solution (i.e. when the corresponding linear system has no
____________-). - Answer-free variable
To determine if the columns of A are linearly independent or not, we study the matrix
equation
A⃗x = ⃗0. This is ___________________. - Answer-equivalent to the linear system with
augmented matrix
A collection of one vector {⃗v} is linearly independent if and only if ⃗v is not the
______________. - Answer-zero vector
A set of two vectors {⃗v1,⃗v2} is linearly dependent exactly when one vector is a multiple
of the other vector.
That is, ⃗v1 and ⃗v2 are _______________________ if and only if neither of ⃗v1 and ⃗v2 is
multiple of the other. - Answer-linearly independent
Geometrically, two vectors are linearly dependent if they lie on the same _______. -
Answer-line
If p > n, there are more variables than equations. it follows that there must be a
________________. - Answer-free variable
QUESTIONS AND ANSWERS
The span of ⃗0 in Rn is ... - Answer-the single vector ⃗0.
The span of a single nonzero vector in R2 and R3 is ... - Answer-a line through ⃗0.
The span of two nonzero vectors ⃗u and ⃗v in R3, with ⃗v ̸= c⃗u, is ... - Answer-a plane
through ⃗0.
An indexed set of vectors S = {⃗v1,⃗v2,...,⃗vp} in Rn is said to be linearly independent (we
may say also that the vectors ⃗v1,...,⃗vp are linearly independent) if the vector equation
x1⃗v1 +x2⃗v2 +···+xp⃗vp =⃗0
has only the __________ solution - Answer-trivial
Recall that the vector equation is equivalent to the homogeneous equation A⃗x = ⃗0 and
he matrix equation A⃗x = ⃗0 has a nontrivial solution if and only if it has a ___________. -
Answer-free variable
The columns of a matrix A are linearly independent exactly when the matrix equation A⃗x
= ⃗0 has
only the trivial solution (i.e. when the corresponding linear system has no
____________-). - Answer-free variable
To determine if the columns of A are linearly independent or not, we study the matrix
equation
A⃗x = ⃗0. This is ___________________. - Answer-equivalent to the linear system with
augmented matrix
A collection of one vector {⃗v} is linearly independent if and only if ⃗v is not the
______________. - Answer-zero vector
A set of two vectors {⃗v1,⃗v2} is linearly dependent exactly when one vector is a multiple
of the other vector.
That is, ⃗v1 and ⃗v2 are _______________________ if and only if neither of ⃗v1 and ⃗v2 is
multiple of the other. - Answer-linearly independent
Geometrically, two vectors are linearly dependent if they lie on the same _______. -
Answer-line
If p > n, there are more variables than equations. it follows that there must be a
________________. - Answer-free variable
