LINEAR ALGEBRA FINAL EXAM
DEFINITIONS QUESTIONS AND
ANSWERS
Orthogonality - Answer-- Two vectors and are orthogonal if they're perpendicular
- If their inner product is 0:
v*w=0
If you change the basis of a symmetric basis what happens? - Answer-It preserves all
lengths and angles
What is Q^t=I called? - Answer-An orthogonal matrix
Are the eigenvalues of eigen vectors of a symmetric orthogonal? - Answer-Yes
Orthogonal Matrix ? - Answer-Q^-1=Q^t
Projection formula with three vectors
can only be used with orthogonal vectors - Answer-P= (U*V/ (V)(V))*V +(U*W/(W)(W))W
Show that QQ^T=I - Answer-V1*V^T...+ VNVN^T
Aei=ei
If U is a subspace of R^n or any vector space with dot products then the orthogonal
complement U perpendicular Of U is? - Answer-U perpendicular :V*U = 0 for all U
D=P^-1 *A*P= P^T A P - Answer-
Rank - Answer-the rank of A is the dimension of the image of A as a subspace of R^m
-The Rank of A is the number of pivot column of A
Nullity - Answer-A dimension of the null space of A the subspace of R^n consisting of x
such that Ax=0
-not changed by RREF
Rank nullity theorem - Answer-Let A be a m*n matrix with Rank A=r and nullity (A)=k
n=r+k
Complete the definition of a rank of a linear transformation L:V->W - Answer-The rank is
the image, the number of pivot columns
Subspace of W
DEFINITIONS QUESTIONS AND
ANSWERS
Orthogonality - Answer-- Two vectors and are orthogonal if they're perpendicular
- If their inner product is 0:
v*w=0
If you change the basis of a symmetric basis what happens? - Answer-It preserves all
lengths and angles
What is Q^t=I called? - Answer-An orthogonal matrix
Are the eigenvalues of eigen vectors of a symmetric orthogonal? - Answer-Yes
Orthogonal Matrix ? - Answer-Q^-1=Q^t
Projection formula with three vectors
can only be used with orthogonal vectors - Answer-P= (U*V/ (V)(V))*V +(U*W/(W)(W))W
Show that QQ^T=I - Answer-V1*V^T...+ VNVN^T
Aei=ei
If U is a subspace of R^n or any vector space with dot products then the orthogonal
complement U perpendicular Of U is? - Answer-U perpendicular :V*U = 0 for all U
D=P^-1 *A*P= P^T A P - Answer-
Rank - Answer-the rank of A is the dimension of the image of A as a subspace of R^m
-The Rank of A is the number of pivot column of A
Nullity - Answer-A dimension of the null space of A the subspace of R^n consisting of x
such that Ax=0
-not changed by RREF
Rank nullity theorem - Answer-Let A be a m*n matrix with Rank A=r and nullity (A)=k
n=r+k
Complete the definition of a rank of a linear transformation L:V->W - Answer-The rank is
the image, the number of pivot columns
Subspace of W
