Similar matrices

Carefully write a proof for the following statement: Similar matrices have the same characteristic polynomials.

Proof : Any n x n singular matrix A must have zero as an eigenvalue.

Carefully write a proof for the following statement: Any n x n singular matrix A must have zero as an eigenvalue.

Linear algebra and differential equations

Determine if the given set constitutes a real vector space. The operations of "multiplication by a number" and "addition" are understood to be the usual operations associated with the elements of the set: The set of all elements of R^3 with first component 0

Differential Equations and Vector Spaces

Determine if the given set constitutes a real vector space. The operations of "multiplication by a number" and "addition" are understood to be the usual operations associated with the elements of the set: The set of all polynomials of degree  2

Geometric Interpretation of the Subspace

Show that the set of all elements of R^2 of the form (a, -a), where a is any real number, is a subspace of R^2. Give a geometric interpretation of the subspace.

Sets : Subspace

Show that the set of all elements of R^2 of the form (1, a), where a is any real number, is not a subspace of R^2.

linear algebra

Show that the set of all elements of R^3 of the form (a + b, -a, 2b), where a and b are any real numbers, is a subspace of R^3. Show that the geometric interpretation of this subspace is a plane and find its equation.

Vector Spaces and Linear Dependence

If S is any finite set of elements of a vector space V that contains the zero element of V, show that S is linearly dependent.

Determine whether the given set of 2x2 matrices are linearly independent

The set of all 2 x 2 real matrices constitutes a real vector space. Determine whether the given set of elements is linearly independent: [ 2 3 ] , [ -1 2 ] ,[ 1 0 ] [ 1 1 ] [ 0 0 ] [ 0 1 ]

Linear Algebra : Wronskian

Compute the Wronskian of the given set of functions, then determine whether the function is linearly dependent or linearly independent: cos ax, sin ax, a  0, x in any interval