X d x example 1 blue function 4 π parenleftbigg sin

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( x )d x. Example 1. blue function = 4 π parenleftbigg sin ( x )+ 1 3 sin (3 x )+ 1 5 sin (5 x )+ 1 7 sin (7 x )+ parenrightbigg - π π 2 π 3 π 4 π Armin Straub [email protected] 1
Example 2. Find the Fourier series of the 2 π -periodic function f ( x ) defined by f ( x )= braceleftbigg 1 , for x ( π, 0) , +1 , for x (0 , π ) . - π π 2 π 3 π 4 π Solution. Note that integraltext 0 2 π and integraltext - π π are the same here. (why?!) a 0 = 1 2 π integraldisplay - π π f ( x )d x =0 a n = 1 π integraldisplay - π π f ( x ) cos ( nx )d x = 1 π bracketleftbigg integraldisplay - π 0 cos ( nx )d x + integraldisplay 0 π cos ( nx )d x bracketrightbigg =0 b n = 1 π integraldisplay - π π f ( x ) sin ( nx )d x = 1 π bracketleftbigg integraldisplay - π 0 sin ( nx )d x + integraldisplay 0 π sin ( nx )d x bracketrightbigg = 2 π bracketleftbiggintegraldisplay 0 π sin ( nx )d x bracketrightbigg = 2 π bracketleftbigg 1 n cos ( nx ) bracketrightbigg 0 π = 2 πn [1 cos ( )] = 2 πn [1 ( 1) n ]= braceleftBigg 4 πn if n is odd 0 if n is even In conclusion, f ( x )= 4 π parenleftbigg sin ( x )+ 1 3 sin (3 x )+ 1 5 sin (5 x )+ 1 7 sin (7 x )+ parenrightbigg . Armin Straub [email protected] 2
Determinants For the next few lectures, all matrices are square! Recall that bracketleftbigg a b c d bracketrightbigg - 1 = 1 ad bc bracketleftbigg d b c a bracketrightbigg . The determinant of a 2 × 2 matrix is det parenleftBig bracketleftbigg a b c d bracketrightbigg parenrightBig = ad bc, a 1 × 1 matrix is det ([ a ])= a. Goal: A is invertible det ( A ) 0 We will write both det parenleftBig bracketleftbigg a b c d bracketrightbigg parenrightBig and vextendsingle vextendsingle vextendsingle vextendsingle a b c d vextendsingle vextendsingle vextendsingle vextendsingle for the determinant. Definition 3. The determinant is characterized by: the normalization det I =1 , and how it is affected by elementary row operations: (replacement) Add one row to a multiple of another row. Does not change the determinant. (interchange) Interchange two rows. Reverses the sign of the determinant. (scaling) Multiply all entries in a row by s . Multiplies the determinant by s . Example 4. Compute vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 1 0 0 0 2 0 0 0 7 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle . Solution. vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 1 0 0 0 2 0 0 0 7 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle R 2 1 2 R 2 2 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 1 0 0 0 1 0 0 0 7 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle R 3 1 7 R 3 14 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 1 0 0 0 1 0 0 0 1 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = 14 Example 5. Compute vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 1 2 3 0 2 4 0 0 7

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