Tutor profile: John H.

How would I use a loop to print out the following in python: * ** *** **** *****

The print syntax depends on if you're using Python 2 or Python 3. Let's assume you're using Python 3. To print a single star you can use the following statement: print("*") To print 5 stars you can do this: print("*"*5) So, to print your output, we can use a nifty loop: for i in range(1,6): print("*"*i) Try that out!

What is the kernel of a linear transformation? And how would I find it if I have a matrix?

Let's call our linear transformation $$L: V \rightarrow W$$. The kernel is a subspace of the domain of the linear transformation. It is the set of all vectors $$v$$ that are sent to zero under the transformation. If you have a matrix for $$L$$, you would put it into reduced row echelon form. If the matrix has a nontrivial kernel there will be rows of zeros in the matrix. Take the nonzero rows and write down their linear equations. There will be free variables since there are less equations than variables. Write down general form solutions with free variables - these will be a basis for your kernel.

What is the derivative of $$3x^2 + cos(x)$$?

We can take the derivative of each individual term that is added together. The derivative of $$3x^2$$ is $$6x$$ from the power rule. The derivative of $$cos(x)$$ is $$-sin(x)$$. So, overall, the derivative is $$6x - sin(x)$$