For each of the following statements, write \"true\" if the statement is true, a

Question

For each of the following statements, write "true" if the statement is true, and write "false" if the statement is false. If A is a 3 times 5 matrix, then the domain of T_A is R^3. If the n times n matrix A is invertible, then the reduced row echelon form of A is the n times n identity matrix. Any matrix, A. can be transformed into reduced row echelon form. rref(A), by a finite sequence of elementary row operations and rre f(A) is unique. The zero vector is orthogonal to every vector except itself. A system of n linear equations in n variables can have at most one solution. There are linear transformations from R' to R that are not matrix transformations. There is only one matrix transformation such that T(-x) = -T(x) for every vector x in R^n. If T: R^n rightarrow R^n is a linear transformation, then T(0) = 0. If A is an n times n matrix such that det(A) notequalto 0. then the system of linear equations denoted by Ax = b has a unique solution. If the n times n matrix A is invertible, then the matrix A^T is also invertible. If A and B are invertible n times n matrices, then AB is invertible and (AB)^-1 = A^-1B^-1 If v is a nonzero vector in R^n then there are exactly two unit vectors that are parallel to v.

Explanation / Answer

a.FALSE The domain is R 5

explaination: The set n is called the domain of A, and m is called the codomain of T.

The notation A: n m says the domain of A is n .

If A is a 3 × 5 matrix and T is a transformation defined by T(x) = Ax, then the domain of T is R 3 .

b.True

explanation:An n x n matrix A is said to be invertible if and only if there is any n x n matrix X such that XA= I and AX= I where I is the n x n identity matrix.

d.true.

Yes, the zero vector is orthogonal to every vector. The only time that two vectors are orthogonal is when their dot product is zero.

e.false

Any system of linear equations can have 0, 1, or infinitely many solutions.

f.FALSE. The converse (every matrix transformation is a linear transformation) is true.

g.False

h. True

If T is a linear transformation, then T(0) = 0 and T(cu + dv) = cT(u) + dT(v) for all scalars c and d. and all vectors u and v in the domain of T.

j.true “the inverse of the transpose of a a matrix is the same as its transpose of the inverse.”

k.true

An n x n matrix (M) is said to be invertible if there is an n x n matrix (C) such that CM= I and MC= I where I is the n x n identity matrix.Basically M-1 M= I and M M-1 = I where M is an invertible matrix and M-1 is the inverse of M.

L.true

The obvious two are a/|a| and -a/|a|, they are both unit and proportional to a. Now assume there is a third unit vector u. Then u=ka and |u| = |k||a|. Since u is also a unit vector, |k||a| = 1 and so |k| = 1/|a| => k = ±1/|v|. Therefore the assumption of a third is a contradiction and thus only the 2 mentioned exist.

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