MATH 410 American River College Linear Algebra Final Exam I need help with these two parts of questions. and i need the answer with all the steps. = M Gmai

MATH 410 American River College Linear Algebra Final Exam I need help with these two parts of questions. and i need the answer with all the steps. = M Gmail
Q Sear
3. Let V R2 with the following rules of vector addition and scalar
multiplication where u = (41, 42) and v= (V1, V2):
ES
m
u v = (U1 + V1 + 1, U2 + V2 + 1)
5 of 271
3
+ Compose
k Ou= (k(U1 + 1) – 1, k(u2 + 1) – 1)
m
Determine whether V with the defined operations is a real vector space
and which axioms are satisfied and which fail. (Hint: 0 may exist but
not be (0,0).)
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4. Give an example of the following or explain why it is impossible.
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(a) A linear transformation from R5 to R2 that is onto.
(b) A linear transformation from R5 to R2 that is one-to-one.
(c) A linear transformation from R3 to R4 that is onto.
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2 A
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(d) A linear transformation from R3 to R4 that is one-to-one.
(e) A linear transformation from R3 to R3 that is one-to-one but not
onto.
(f) A 4 x 3 matrix with nullity zero.
(g) A 3 x 4 matrix with nullity zero.
(h) A matrix A with Null(A) = span{(1,1,0,0), (0,1,1,0)}.
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158
1. Let T:R” + RM be a linear transformation and A the standard matrix
of T.
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ton
=
2 A
(a) Let BN = {V1, …,Vr} be a basis for ker(T) (i.e. Null(A)). We ex-
tend Bn to a basis for R” and denote it by B {V1, … , Vr, Ur+1, …, Un}.
Show the set BR = {T(ur+1), …,T(un)} is a basis for range(T)
(i.e. col(A)). Conclude that dim(col(A)) = n – dim(Null(A)).
(b) Show that row(A) = Null(A)+ (i.e. ker(T)-).
(c) We have established that for any subspace W of R” (or of
any
finite
dimensional vector space V), dim(W)+dim(W+) = dim(R”) =n
(or dim(W)+dim(W+) = dim(V)). Conclude that dim(col(A)) =
dim(row(A))
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EC Let V be a vector space and let W1 and W2 be subspaces. We have
established that the intersection Win W2 is also a subspace of V and
that it is not true that the union WUW2 is also a subspace of V. For 5
points extra credit, construct the “smallest” subspace of V containing
WUW2 and prove that the subspace you constructed is a subspace of
V and that it is the “smallest” such subspace (i.e. if U is any subspace
containing WLUW2, then it also contains the subspace you constructed).
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1. Find a subset of {V1, V2, V3, V4, V5} that forms a basis for span{V1, V2, V3, V4, V5}
and write the remaining vectors as a linear combination of those where
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31.
+ Compose
V1 = (1,2,1,1), V2 = (-2, -4,-2, -2), V3 = (-1,0,1,1),
V4 (0,1,-1,-1), V5 = (0,3,1,1)
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2. Find the transition matrix PB_B’ from the basis B to the basis B’ where
B = {(-1,1,2), (1,-1,1), (0, 1, 1)} and B’ = {(2,1,1), (2,0,1), (1,1, -1)}.
Then apply the matrix to find (v)B’ where (v)B = (-3, 2,9).
3. Let V R2 with the following rules of vector addition and scalar
multiplication where u = (U1, U2) and v= (V1, V2):
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u v = (U1 + v1 +1, 42 + 02 + 1)
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2A
k Ou= (k(u1 + 1) – 1, k(u2 + 1) – 1)
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Determine whether V with the defined operations is a real vector space
and which axioms are satisfied and which fail. (Hint: 0 may exist but
not be (0,0).)
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4. Give an example of the following or explain why it is impossible.
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(a) A linear transformation from R5 to R2 that is onto.
(b) A linear transformation from R5 to R2 that is one-to-one.
(c) A linear transformation from R3 to R4 that is onto.
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Q Sear
3. Let V R2 with the following rules of vector addition and scalar
multiplication where u = (41, 42) and v= (V1, V2):
ES
m
u v = (U1 + V1 + 1, U2 + V2 + 1)
5 of 271
3
+ Compose
k Ou= (k(U1 + 1) – 1, k(u2 + 1) – 1)
m
Determine whether V with the defined operations is a real vector space
and which axioms are satisfied and which fail. (Hint: 0 may exist but
not be (0,0).)
2 기
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158
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Ha
4. Give an example of the following or explain why it is impossible.
| 31 AM (10 minutes ago)
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to
Sent
(a) A linear transformation from R5 to R2 that is onto.
(b) A linear transformation from R5 to R2 that is one-to-one.
(c) A linear transformation from R3 to R4 that is onto.
Drafts
2 A
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(d) A linear transformation from R3 to R4 that is one-to-one.
(e) A linear transformation from R3 to R3 that is one-to-one but not
onto.
(f) A 4 x 3 matrix with nullity zero.
(g) A 3 x 4 matrix with nullity zero.
(h) A matrix A with Null(A) = span{(1,1,0,0), (0,1,1,0)}.
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