If the statement is true, prove it. Otherwise give a counter example. a)If V=C3 and W1={(z1,z2,z2)∈C3:z1,z2∈C},...

If the statement is true, prove it. Otherwise give a counter example.

a)If V=C3 and W1={(z1,z2,z2)∈C3:z1,z2∈C}, W2={(0,z,0)∈C3:z∈C}, then V=W1⊕W2.

b)If Vis a vector space and W1, W2 are subspaces of V, then W1∪W2 is also a subspace of V.

c)If T:V→V is a linear operator, then Ker(T) and Range(T) are invariant under T.

d)Let T:V→V be a linear operator. If Ker(T)∩Range(T) ={0}, then V=Ker(T)⊕Range(T).

e)If T1,T2:V→V are linear operators such that T1T2=T2T1, and λ2 is an eigenvalue of T2, then Ker(T2−λ2I) is invariant under T1.

Expert Solution

ekkarill92 answered 2 years ago

Prove that the cross ratio of four complex numbers z1, z2, z3, z4 is real if...

Prove that the cross ratio of four complex numbers z1, z2, z3, z4 is real if and only if the points z1, z2, z3, z4 lie on a line or a circle. Then, compute the cross ratio of 1+√ 3, 1−3i, −1 − i and 1 + i and determine whether they lie on a line, a circle, or neither.

Suppose that W1 and W2 are subspaces of V and dimW1<dimW2. Prove that there is a...

Suppose that W1 and W2 are subspaces of V and dimW1<dimW2. Prove that there is a nonzero vector in W2 which is orthogonal to all vectors in W1.

(a) Let U={(z1, z2, z3, z4, z5)∈C: 6z1=z2, z3+ 2z4+ 3z5= 0}. Check if U is...

(a) Let U={(z1, z2, z3, z4, z5)∈C: 6z1=z2, z3+ 2z4+ 3z5= 0}. Check if U is a subspace. If U is a subspace, then find a basis which span U.

For each statement below circle TRUE if the statement is true, otherwise circle FALSE.

For each statement below circle TRUE if the statement is true, otherwise circle FALSE. A hypothesis test uses data from a sample to assess a claim about a population. A randomization distribution will be centered at the value used in the null hypothesis. “Failing to reject the null hypothesis” and “accepting the null hypothesis” mean the same thing in a hypothesis test’s conclusion. The p-value is the probability, assuming the alternative hypothesis is true, of obtaining a sample statistic as...

(need in frame -c) In the given code, Will all the assert statements prove true? Give...

(need in frame -c) In the given code, Will all the assert statements prove true? Give reason for proved true or false for each assert statement. [3 M] [CO2] int main(){ int i = 1; int x = 33; /*@ loop invariant 1 <= i <= 19; loop assigns i; loop variant 20 - i; */ //@assert i == 1; while(i < 20){ ++i; if(i == 20) break ; } //@assert i == 20; //@assert x == 33; } frama-c...

Determine whether each statement is true or false. If it is true, prove it. If it...

Determine whether each statement is true or false. If it is true, prove it. If it is false, give a counterexample. a) For every function f : X → Y and all A ⊆ X, we have f^−1 [f[A]] = A. (b) For every function f : X → Y and all A ⊆ X, we have f[X \ A] = Y \ f[A]. (c) For every function f : X → Y and all A, B ⊆ Y ,...

In each case below either show that the statement is True or give an example showing...

In each case below either show that the statement is True or give an example showing that it is False. (V) If {x1, x2, . . . xk, xk+1, . . . xn} is a basis of R n and U = span{x1, . . . xk} and V = span{xk+1 . . . xn}, then U ∩ V = {0}.

1. Determine if the following statements are true or false. If a statement is true, prove...

1. Determine if the following statements are true or false. If a statement is true, prove it in general, If a statement is false, provide a specific counterexample. Let V and W be finite-dimensional vector spaces over field F, and let φ: V → W be a linear transformation. A) If φ is injective, then dim(V) ≤ dim(W). B) If dim(V) ≤ dim(W), then φ is injective. C) If φ is surjective, then dim(V) ≥ dim(W). D) If dim(V) ≥...

NOTE- If it is true, you need to prove it and If it is false, give...

NOTE- If it is true, you need to prove it and If it is false, give a counterexample f : [a, b] → R is continuous and in the open interval (a,b) differentiable. a) If f(a) ≥ f(b), then exists a ξ ∈ (a,b) with f′(ξ) ≤ 0.(TRUE or FALSE?) b) If f is reversable, then f −1 differentiable. (TRUE or FALSE?) c) If f ′ is limited, then f is lipschitz. (TRUE or FALSE?)

NOTE- If it is true, you need to prove it and If it is false, give...

NOTE- If it is true, you need to prove it and If it is false, give a counterexample f : [a, b] → R is continuous and in the open interval (a,b) differentiable. a) If f(a) ≥ f(b), then exists a ξ ∈ (a,b) with f′(ξ) ≤ 0. (TRUE or FALSE?) b) If f is reversable, then f −1 differentiable. (TRUE or FALSE?) c) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?)

Question

If the statement is true, prove it. Otherwise give a counter example. a)If V=C3 and W1={(z1,z2,z2)∈C3:z1,z2∈C},...

Solutions

Expert Solution

Related Solutions

Prove that the cross ratio of four complex numbers z1, z2, z3, z4 is real if...

Suppose that W1 and W2 are subspaces of V and dimW1<dimW2. Prove that there is a...

(a) Let U={(z1, z2, z3, z4, z5)∈C: 6z1=z2, z3+ 2z4+ 3z5= 0}. Check if U is...

For each statement below circle TRUE if the statement is true, otherwise circle FALSE.

(need in frame -c) In the given code, Will all the assert statements prove true? Give...

Determine whether each statement is true or false. If it is true, prove it. If it...

In each case below either show that the statement is True or give an example showing...

1. Determine if the following statements are true or false. If a statement is true, prove...

NOTE- If it is true, you need to prove it and If it is false, give...

NOTE- If it is true, you need to prove it and If it is false, give...