By Denton, Waldron

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**Example text**

In quantum physics, vector spaces over C describe all possible states a system of particles can have. For example, λ V = : λ, µ ∈ C µ describes states of an electron, where 1 0 describes spin “down”. Other states, like describes spin “up” and i −i 0 1 are permissible, since the base field is the complex numbers. Complex numbers are extremely useful because of a special property that they enjoy: every polynomial over the complex numbers factors into a product of linear polynomials. For example, the polynomial x2 + 1 doesn’t factor over the real numbers, but over the complex numbers it factors into (x + i)(x − i).

S, then M N = L where L = ( ij ) for i = i, . . , r; j = 1, . . , s is given by k i j mip npj . = p=1 This rule obeys linearity. Notice that in order for the multiplication to make sense, the columns and rows must match. For an r × k matrix M and an s × m matrix N , then to make the product M N we must have k = s. Likewise, for the product N M , it is required that m = r. A common shorthand for keeping track of the sizes of the matrices involved in a given product is: r×k × k×m = r×m 56 Example Multiplying a (3 × 1) matrix and a (1 × 2) matrix yields a (3 × 2) matrix.

Does not matter, we see that the two computations give exactly the same answer. Of course, you should complain that we took a special choice of α and β. Actually, to take care of this we only need to check that f (αX) = αf (X). It is your job to explain this in Review Question 1 Later we will show that matrix multiplication is always linear. Then we will know that: M (αX + βY ) = αM X + βM Y Then the two equations M X = V and X = X0 + µ1 Y1 + µ2 Y2 together say that: M X 0 + µ1 M Y 1 + µ2 M Y 2 = V for any µ1 , µ2 ∈ R.