Please, show me. Let x and y be vectors in R4; let v and w be vectors in R4, and let A be a 4 times 3 matrix. Determine whether each of the following quantities are a scalar, a vector (in which case you should say if it belongs to R3 or R4), a matrix (in which case you should give its size), or nonsense. For the purposes of this problem, matrices with a single column should be called vectors. (v middot w) Ax A((x middot y)Aw middot y) dist(x, x) ||2y + x|| ||y||y ||w||x dist(w, 2v)A dist(x, w)y middot x Solution (a) (v.w) is a scalar as it is the dot product. A is 4*3 matrix and x is 4*1 matrix (as it is a vector). Thus Ax is an invalid multiplication. Nonsense (b) x.y is a scalar. A is a 4*3 matrix and w is 3*1 matrix. Thus Aw is a 4*1 matrix. Aw.y is a scalar. Thus it is scalar*A. Matrix 4*3 (c) distance between two vector is a scalar. Scalar. (d) ||2y+x|| is a norm. Thus it is a scalar. (e) ||y|| is a norm and thus a scalar. ||y||y is a vector belonging to R^4. Vector in R^4 (f) ||w|| is a norm and thus a scalar. ||w||x is a vector belonging to R^4. Vector in R^4 (g) dist(w,2v) is a scalar and scalar*A is a matrix. Matrix with dimension 4*3. (h) dist(x,w) is a scalar. dist(x,w)y is a vector and the dot product is a scalar. Scalar.

1. Please, show me. Let x and y be vectors in R4; let v and w be vectors in R4, and let A be a 4
times 3 matrix. Determine whether each of the following quantities are a scalar, a vector (in
which case you should say if it belongs to R3 or R4), a matrix (in which case you should give its
size), or nonsense. For the purposes of this problem, matrices with a single column should be
called vectors. (v middot w) Ax A((x middot y)Aw middot y) dist(x, x) ||2y + x|| ||y||y ||w||x
dist(w, 2v)A dist(x, w)y middot x
Solution
(a)
(v.w) is a scalar as it is the dot product.
A is 4*3 matrix and x is 4*1 matrix (as it is a vector).
Thus Ax is an invalid multiplication.
Nonsense
(b)
x.y is a scalar. A is a 4*3 matrix and w is 3*1 matrix.
Thus Aw is a 4*1 matrix. Aw.y is a scalar.
Thus it is scalar*A.
Matrix 4*3

2. (c)
distance between two vector is a scalar.
Scalar.
(d)
||2y+x|| is a norm.
Thus it is a scalar.
(e)
||y|| is a norm and thus a scalar.
||y||y is a vector belonging to R^4.
Vector in R^4
(f)
||w|| is a norm and thus a scalar.
||w||x is a vector belonging to R^4.
Vector in R^4
(g)
dist(w,2v) is a scalar and scalar*A is a matrix.
Matrix with dimension 4*3.
(h)
dist(x,w) is a scalar.
dist(x,w)y is a vector and the dot product is a scalar.
Scalar