Week 10 Q15 on echelon reduction

Hey, just writing up some details to clarify this question. Let me know if it makes sense! 

echelon reduction example with span (Ch4 Al;g Q25)

 

Here is a run through of that problem we did in class, and thanks to that student who had a go on the board!! You will notice the numbers below are the same as yours so you got it right! we just drew the wrong conclusion as we expected a solution, but the answer is that there is NO solution and so the vector is not in the span. :-)

Week 10 Alg Tutorial - solving systems of linear equations by row reduction etc

Three planes can intersect in various ways: one point, one line, one plane, not at all. 

This one has a unique solution.

Geometric interpretation - 3 planes meeting in the point $$(-2,1,2)$$.

(Check this point satisfies all 3 equations)

Solution is all the points on a line in parametric form.

Geometric interpretation: 3 planes intersect in a line.

The solution is all the points on the plane shown.

Geometric interpretation - The 3 equations are all the same plane!!

The is no solution. No point lies on all 3 planes.

Week 10 Calc tutorial

 

Basic complex numbers calculations

 

Example of Inverse Function Theorem with $f(x)=x(x^2-25)^2$

 

Basic calculation with complex numbers, example (class test 2014)

 

Basic complex number calculations (class test 2014)

 

Find roots and factorise $z^6+64$ (class test 2014)

 

Factorise into real linear and quadratic factors, $p(z)=z^5-32$, (Alg Class test 2014) Q3

 

Find the complex square roots (Alg Class test 2014)

 

Factorise $p(z)=z^5+z^4-8z^3-28z^2+16z+96$ into linear factors.

 The original question was that given $-2-2i$ is a root and that $3$ other roots were real, factorise into linear factors. Since $p(z)$ has real coefficients this means the conjugate root is also a root, so both $-2-2i$ and $-2+2i$ are roots. 

Here we assume that there are $3$ real roots and go from there.

First example of solving linear system - with matrices in mind

 

Chain Rule/Composite Function Rule and Notation.