MA 265 EXAM 2 QUESTIONS WITH COMPLETE
SOLUTIONS 100% SOLVED
Vector Space - ANSWER is a collection of objects called vectors such that
a) any vector can be scaled by a scalar
b)there is a zero vector such that v +0 = v for every vector
c)any two vectors v and w in V can be added
Example of Vector Spaces - ANSWER 1. IR^n with usual addition and scalar multiplication
2. Let V be the set of functions IR --> IR Then define f +g to be the function whose value at x
is cf(x).0 is the zero function. Then, the polynomials of degree less than or equal to n.
Existence and Uniqueness Theorem - ANSWER states that a linear system is consistent (has a
solution) if and only if the rightmost column of the augmented matrix is is not a pivot
column. That is, a linear system is consistent if and only if an echelon form of the augmented
matrix has no row of the form [0 * * * 0 b] where b is nonzero
Scalar multiplication does not change - ANSWER the degree of the polynomial
Subspace (H) - ANSWER of a vector space is a collection of vectors which satisfy the usual
properites
,Null Space - ANSWER Set of all solution to Ax = 0
Finding the Null Space - ANSWER - Add a column of 0s
- Row reduce
- Find which rows are not pivot columns
-Find the equations for each of the variables
- Set them to the free variables
-Those vectors are you null space vectors
Column Space - ANSWER Ax = b
is the span of its column vectors
Finding the Column Space - ANSWER - row reduce
- see the rows with the pivot columns
- look a the original matrix and that is the basis for your columns space
Row Space - ANSWER row space of a matrix A is defined to be the Col(A^t)
If A is a m x n matrix. The row space is a subspace of IR^n
Linear Transformation - ANSWER T: V--> W is a function which assigns to every vector v in V a
vector Tv in W such that
, a) T(v+w) = Tv =Tw
b) T(cv) = cTv
For any linear transformation T we have two important invariants: - ANSWER 1. Kernel of T -->
collection of vectors v in V such that Tv = 0
2. Range of T--> collection of vectors w in W such that w = Tv for some vector v in V. This is a
subspace of w.
An indexed set of vectors in V is said to be linearly independent if - ANSWER c1v1 + c2v2 +
cpvp = 0
has only one solution.
{v11, v2, ... vp} in V is linearly independent if - ANSWER c1v1 + c2v2 +cpvp = 0 has only the
trival solution
c1 = c2 = cp = 0
An indexed set of 2 or more vectors w/ v1 = 0 is linearly dependent if and only if - ANSWER
some vJ (j > 1) is a linear combination of the preceding vectors
What does it mean for a set to be a basis of some subspace H of a vector space? - ANSWER
Set is linearly independent
SOLUTIONS 100% SOLVED
Vector Space - ANSWER is a collection of objects called vectors such that
a) any vector can be scaled by a scalar
b)there is a zero vector such that v +0 = v for every vector
c)any two vectors v and w in V can be added
Example of Vector Spaces - ANSWER 1. IR^n with usual addition and scalar multiplication
2. Let V be the set of functions IR --> IR Then define f +g to be the function whose value at x
is cf(x).0 is the zero function. Then, the polynomials of degree less than or equal to n.
Existence and Uniqueness Theorem - ANSWER states that a linear system is consistent (has a
solution) if and only if the rightmost column of the augmented matrix is is not a pivot
column. That is, a linear system is consistent if and only if an echelon form of the augmented
matrix has no row of the form [0 * * * 0 b] where b is nonzero
Scalar multiplication does not change - ANSWER the degree of the polynomial
Subspace (H) - ANSWER of a vector space is a collection of vectors which satisfy the usual
properites
,Null Space - ANSWER Set of all solution to Ax = 0
Finding the Null Space - ANSWER - Add a column of 0s
- Row reduce
- Find which rows are not pivot columns
-Find the equations for each of the variables
- Set them to the free variables
-Those vectors are you null space vectors
Column Space - ANSWER Ax = b
is the span of its column vectors
Finding the Column Space - ANSWER - row reduce
- see the rows with the pivot columns
- look a the original matrix and that is the basis for your columns space
Row Space - ANSWER row space of a matrix A is defined to be the Col(A^t)
If A is a m x n matrix. The row space is a subspace of IR^n
Linear Transformation - ANSWER T: V--> W is a function which assigns to every vector v in V a
vector Tv in W such that
, a) T(v+w) = Tv =Tw
b) T(cv) = cTv
For any linear transformation T we have two important invariants: - ANSWER 1. Kernel of T -->
collection of vectors v in V such that Tv = 0
2. Range of T--> collection of vectors w in W such that w = Tv for some vector v in V. This is a
subspace of w.
An indexed set of vectors in V is said to be linearly independent if - ANSWER c1v1 + c2v2 +
cpvp = 0
has only one solution.
{v11, v2, ... vp} in V is linearly independent if - ANSWER c1v1 + c2v2 +cpvp = 0 has only the
trival solution
c1 = c2 = cp = 0
An indexed set of 2 or more vectors w/ v1 = 0 is linearly dependent if and only if - ANSWER
some vJ (j > 1) is a linear combination of the preceding vectors
What does it mean for a set to be a basis of some subspace H of a vector space? - ANSWER
Set is linearly independent
