It should look like a parabola with the part below x-axis cut off. the left side starts from x=-1, opening upward & to the left, the right side starts from x=2, opening upward and to the right.
Step-by-step explanation:
Name the total of pens purchased as x and the pencils as y... Make two equations and solve them. I also search for the pencils purchased in addition
This is easy once you look at it you just have to move the decimal 7 places and you'll get 24,000,000
A vector space is a subspace of a vector space if
- is non-empty,
- for any two vectors we have , and
- for any scalar and we have .
It's easy to show the first condition is met by all the sets in parts (a-g).
(a) is a subspace of because adding any 2x2 diagonal matrices together, or multiplying one by some scalar, gives another diagonal matrix.
(b) and (c) are also subspaces for the same reasons.
(d) is not a subspace because because this set of matrices does not contain the zero matrix.
(e), however, is a subspace. Any linear combination of matrices in this set always yields a matrix with 0 in row 1, column 1 entry.
(f) is a subspace. A symmetric matrix is one of the form
Adding two symmetric matrices gives another symmetric matrix:
(g) is not a subspace. Consider the matrices
Both matrices have determinant 0, but their sum is the identity matrix with determinant 1.