Name: Score:
43 Multiple choice questions
Term
Find a unit vector in the direction of v
yes, if det does not equal 0
vector/length of vector
dimension of column space
basis for v has p elements1 of 43
Term
(T/F) For a square matrix A, suppose Ax=b has no solutions. Then A^T is not invertible
True
False2 of 43
Term
(T/F) If A and B are n x n matrices with detA=2 and detB=3, then det(A+B) = 5
True
False3 of 43
Term
(T/F) If A is an n x n matrix and B is obtained by switching two of its rows, then detA=detB.
True
False4 of 43 Term
Comput e the orthogonal projection of u onto v
1. contains zero vector
2. closed under addition
3. closed under multiplication b y a scalar
((uv)/(vv))V
1. Span of vectors=V
2. linearly independent
vector/length of vector5 of 43
Term
Determine if linearl y independent
Yes, if det does not equal 0
-Contains zero vector
-Closed under addition
-Closed under mult. by a scalar
Must have a free variabl e, columns not multiples of each other
Pivot columns (linear independent columns ) of original matrix6 of 43
Term
consis tent
A system that has one or infinit ely many solutions
Dimension of the nul l space (free variabl es)
t(2 0 - 1) sinc e v is in R^3 and sp ansW, W is a subspace
A system that does not have a solution7 of 43 Term
Find the eigen vector
Solve for the eigen values. Set the piv ot point s equal to pivot #-lambda, plug in the
eigen value for lambda. Set the ma trix equal to 0. Solve for the vector in terms of the free
variabl e.
Set the piv ots equal to pivot #-lambda. Find the determinant . Solve for lambda.
1. A = vector of 1, a
2. b = b
3. solve A^TA and A^Tb
4. A^TA (B_0 , B_1) = A^Tb, solve for B_0 and B_1
5. Answer shoul d be in form y = B_0 + B_1( x)
NulA = set of all sols to Ax=0
ColA=set of all linear combina tions of the columns of A(span of cols)
RowA= set of all linear combo s of the r ows of A(span of rows)8 of 43
Term
Determine if invertibl e
1. span of vectors=v 2. linearly independent
Dimension of the nul l space (free variabl es)
Yes, if det does not equal 0
Yes, if det does not equal 59 of 43 Term
Define Span(set of vectors)
NulA = set of all sols to Ax=0
ColA=set of all linear combina tions of the columns of A(span of cols)
RowA= set of all linear combo s of the r ows of A(span of rows)
1. Set Ax=0
2. Solve for weight s
3. vector in terms of free variabl e is a basis
All possibl e linear combina tions, all vectors that can be writt en in the form c1v1+...+cp vp
wher e ci's are scalars.
Solve for the eigen values. Set the piv ot point s equal to pivot #-lambda, plug in the
eigen value for lambda. Set the ma trix equal to 0. Solve for the vector in terms of the free
variabl e.10 of 43
Term
nullity
Basis for v has p element s
The r ow x column ma tch
Yes, if det does not equal 0
Dimension of the nul l space (free variabl es)11 of 43
43 Multiple choice questions
Term
Find a unit vector in the direction of v
yes, if det does not equal 0
vector/length of vector
dimension of column space
basis for v has p elements1 of 43
Term
(T/F) For a square matrix A, suppose Ax=b has no solutions. Then A^T is not invertible
True
False2 of 43
Term
(T/F) If A and B are n x n matrices with detA=2 and detB=3, then det(A+B) = 5
True
False3 of 43
Term
(T/F) If A is an n x n matrix and B is obtained by switching two of its rows, then detA=detB.
True
False4 of 43 Term
Comput e the orthogonal projection of u onto v
1. contains zero vector
2. closed under addition
3. closed under multiplication b y a scalar
((uv)/(vv))V
1. Span of vectors=V
2. linearly independent
vector/length of vector5 of 43
Term
Determine if linearl y independent
Yes, if det does not equal 0
-Contains zero vector
-Closed under addition
-Closed under mult. by a scalar
Must have a free variabl e, columns not multiples of each other
Pivot columns (linear independent columns ) of original matrix6 of 43
Term
consis tent
A system that has one or infinit ely many solutions
Dimension of the nul l space (free variabl es)
t(2 0 - 1) sinc e v is in R^3 and sp ansW, W is a subspace
A system that does not have a solution7 of 43 Term
Find the eigen vector
Solve for the eigen values. Set the piv ot point s equal to pivot #-lambda, plug in the
eigen value for lambda. Set the ma trix equal to 0. Solve for the vector in terms of the free
variabl e.
Set the piv ots equal to pivot #-lambda. Find the determinant . Solve for lambda.
1. A = vector of 1, a
2. b = b
3. solve A^TA and A^Tb
4. A^TA (B_0 , B_1) = A^Tb, solve for B_0 and B_1
5. Answer shoul d be in form y = B_0 + B_1( x)
NulA = set of all sols to Ax=0
ColA=set of all linear combina tions of the columns of A(span of cols)
RowA= set of all linear combo s of the r ows of A(span of rows)8 of 43
Term
Determine if invertibl e
1. span of vectors=v 2. linearly independent
Dimension of the nul l space (free variabl es)
Yes, if det does not equal 0
Yes, if det does not equal 59 of 43 Term
Define Span(set of vectors)
NulA = set of all sols to Ax=0
ColA=set of all linear combina tions of the columns of A(span of cols)
RowA= set of all linear combo s of the r ows of A(span of rows)
1. Set Ax=0
2. Solve for weight s
3. vector in terms of free variabl e is a basis
All possibl e linear combina tions, all vectors that can be writt en in the form c1v1+...+cp vp
wher e ci's are scalars.
Solve for the eigen values. Set the piv ot point s equal to pivot #-lambda, plug in the
eigen value for lambda. Set the ma trix equal to 0. Solve for the vector in terms of the free
variabl e.10 of 43
Term
nullity
Basis for v has p element s
The r ow x column ma tch
Yes, if det does not equal 0
Dimension of the nul l space (free variabl es)11 of 43
