Question

In: Advanced Math

Let x ∈ Rn be any nonzero vector. Let W ⊂ Rnxn consist of all matrices...

Let x ∈ Rn be any nonzero vector. Let W ⊂ Rnxn
consist of all matrices A such that Ax = 0. Show that W is a subspace and find its dimension.

Solutions

Expert Solution


Related Solutions

Let W be a subspace of Rn. Prove that W⊥ is also a subspace of Rn.
Let W be a subspace of Rn. Prove that W⊥ is also a subspace of Rn.
2) Let v, w, and x be vectors in Rn. a) If v is the zero...
2) Let v, w, and x be vectors in Rn. a) If v is the zero vector, what geometric object represents all linear combinations of v? b) Same question as a), except now for a nonzero v. c) Same question as a) except now for nonzero vectors v and w (be care- ful!). d) Same question as a) except now for nonzero vectors v, w, and x (be extra careful!).
Let x = (x1,...,xn) ∼ N(0,In) be a MVN random vector in Rn. (a) Let U...
Let x = (x1,...,xn) ∼ N(0,In) be a MVN random vector in Rn. (a) Let U ∈ Rn×n be an orthogonal matrix (UTU = UUT = In) and nd the distribution of UTx. Let y = (y1,...,yn) ∼ N(0,Σ) be a MVN random vector in Rn. Let Σ = UΛUT be the spectral decomposition of Σ. (b) Someone claims that the diagonal elements of Λ are nonnegative. Is that true? (c) Let z = UTy and nd the distribution of...
Let x ∈ R3 be nonzero and let A be the matrix whose columns are x,...
Let x ∈ R3 be nonzero and let A be the matrix whose columns are x, 2x, 3x in this order. Show that x is an eigenvector of A and find a basis for the null space of A.
Let V be the vector space of 2 × 2 real matrices and let P2 be...
Let V be the vector space of 2 × 2 real matrices and let P2 be the vector space of polynomials of degree less than or equal to 2. Write down a linear transformation T : V ? P2 with rank 2. You do not need to prove that the function you write down is a linear transformation, but you may want to check this yourself. You do, however, need to prove that your transformation has rank 2.
I have a question in that if v is s any nonzero vector, and v is...
I have a question in that if v is s any nonzero vector, and v is positioned with its initial point at the origin, then the terminal points of all scalar multiples of v will occur at all the points on a straight line through the origin. But if we want to find two vectors, let's say m and n, that are parallel to each other, we need to determine whether they are multiples of each other. So my question...
Questionnnnnnn a. Let V and W be vector spaces and T : V → W a...
Questionnnnnnn a. Let V and W be vector spaces and T : V → W a linear transformation. If {T(v1), . . . T(vn)} is linearly independent in W, show that {v1, . . . vn} is linearly independent in V . b. Define similar matrices c Let A1, A2 and A3 be n × n matrices. Show that if A1 is similar to A2 and A2 is similar to A3, then A1 is similar to A3. d. Show that...
Let Vand W be vector spaces over F, and let B( V, W) be the set...
Let Vand W be vector spaces over F, and let B( V, W) be the set of all bilinear forms f: V x W ~ F. Show that B( V, W) is a subspace of the vector space of functions 31'( V x W). Prove that the dual space B( V, W)* satisfies the definition of tensor product, with respect to the bilinear mapping b: V x W -> B( V, W)* defined by b(v, w)(f) =f(v, w), f E...
Let R be the ring of all 2 x 2 real matrices. A. Assume that A...
Let R be the ring of all 2 x 2 real matrices. A. Assume that A is an element of R such that AB=BA for all B elements of R. Prove that A is a scalar multiple of the identity matrix. B. Prove that {0} and R are the only two ideals. Hint: Use the Matrices E11, E12, E21, E22.
Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation....
Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation. We say a linear transformation ?:?→? is a left inverse of ? if ST=I_v, where ?_v denotes the identity transformation on ?. We say a linear transformation ?:?→? is a right inverse of ? if ??=?_w, where ?_w denotes the identity transformation on ?. Finally, we say a linear transformation ?:?→? is an inverse of ? if it is both a left and right...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT