Is A Diagonal Matrix A Subspace?

Is the set of all 2×2 diagonal matrices a subspace?

(a) The set of all 2 × 2 diagonal matrices is a subspace of R2×2, since a scalar multiple of a diagonal matrix is diagonal and the sum of two diagonal matrices is diagonal..

What does diagonal matrix mean?

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. An example of a 2-by-2 diagonal matrix is , while an example of a 3-by-3 diagonal matrix is.

Is an upper triangular matrix a subspace?

Answer: The sum of two upper triangular matrices is obviously an upper triangular matrix and the product of an upper triangular matrix by a real number is an upper triangular matrix. It means that the set of upper triangular matrices is closed with respect to linear operations and is a subspace.

What is lower matrix?

A lower triangular matrix is one which contains all its non-zero elements in and below its main diagonal, as in (1.8).

What is diagonal matrix example?

Any given square matrix where all the elements are zero except for the elements that are present diagonally is called a diagonal matrix. … There are many other matrices other than the Diagonal Matrix, such as symmetric matrix, antisymmetric, diagonal matrix, etc. Example of a Diagonal Matrix = 2.

Is a matrix a diagonal?

A diagonal matrix is a square matrix whose off-diagonal entries are all equal to zero. A diagonal matrix is at the same time: upper triangular; lower triangular.

How do you know if its a subspace?

In other words, to test if a set is a subspace of a Vector Space, you only need to check if it closed under addition and scalar multiplication. Easy! ex. Test whether or not the plane 2x + 4y + 3z = 0 is a subspace of R3.

What is a subspace of a matrix?

In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace, when the context serves to distinguish it from other types of subspaces.

Is r3 a subspace of r4?

It is rare to show that something is a vector space using the defining properties. … And we already know that P2 is a vector space, so it is a subspace of P3. However, R2 is not a subspace of R3, since the elements of R2 have exactly two entries, while the elements of R3 have exactly three entries.

Does a subspace have to contain the zero vector?

Every vector space, and hence, every subspace of a vector space, contains the zero vector (by definition), and every subspace therefore has at least one subspace: … It is closed under vector addition (with itself), and it is closed under scalar multiplication: any scalar times the zero vector is the zero vector.

Is WA subspace of V?

Let V be a vector space over a field F and let W ⊆ V . W is a subspace if W itself is a vector space under the same field F and the same operations. There are two sets of tests to see if W is a subspace of V .

What does a diagonal matrix look like?

A diagonal matrix is defined as a square matrix in which all off-diagonal entries are zero. (Note that a diagonal matrix is necessarily symmetric.) Entries on the main diagonal may or may not be zero. If all entries on the main diagonal are equal scalars, then the diagonal matrix is called a scalar matrix.

Do diagonal matrices commute with everything?

Examples. The unit matrix commutes with all matrices. Every diagonal matrix commutes with all other diagonal matrices.