Are eigenvalues on the linear algebra midterm?

Usually they are the next exam. This midterm is row reduction, inverses, determinants, span, basis, and rank-nullity. Check whether the characteristic polynomial has been assigned.

Can I use a calculator to row-reduce?

Often no, beyond basic arithmetic. Many syllabi ban rref on a calculator or computer. Practice the hand steps even if a device is allowed for numbers.

What is the one theorem worth writing out?

The slice of the invertible matrix theorem your class proved: pivots in every row and column, only the trivial solution to Ax = 0, determinant not zero, and the columns forming a basis. Use their wording.

Linear AlgebraMidterm6 topics

Linear Algebra midterm study sheet

The linear algebra midterm is Gaussian elimination plus the vocabulary that explains it. You row-reduce, decide when a matrix is invertible, compute a determinant by the method your class chose, and say what span, linear independence, and basis mean for a small set of vectors. Eigenvalues, diagonalization, and orthogonality are the usual next exam, after this vocabulary is in place.

Whether it is a formula sheet, an aid sheet or a crib sheet in your course, it is one page of definitions, theorems and the computations you must do by hand.

Make this from your notes Read the reference sheet

Preview of the Linear Algebra reference sheet — the reference sheet for this exam

The scope

What this exam usually covers

Write the theorem as a checklist you can apply to a matrix, not as a paragraph you cannot use under a time limit.

Row reduction To row-echelon and to reduced row-echelon form. Free variables versus pivot variables, and the solution set written in parametric form. Arithmetic errors are the grade; the steps are not optional.
Matrix algebra When AB is defined, that multiplication is not commutative, and how to read a linear system as Ax = b. The inverse, when it exists, from the augmented identity.
The invertible matrix theorem, sliced The handful of equivalent facts your instructor listed: full pivots, trivial null space, columns span R n, determinant nonzero. Not the twenty-line poster unless they required it.
Determinants The method you practiced: cofactor expansion along a row of zeros, or row operations with the sign and scale rules. One 3 by 3 is enough on the sheet as a pattern.
Vector spaces Span, linear independence, basis, dimension, and a subspace test. Column space versus null space, computed from a reduced matrix.
Rank-nullity Rank plus nullity equals the number of columns. Use it to check a basis you already found, not as a substitute for row reduction.

Usually later

Eigenvalues, eigenvectors, diagonalization, Gram-Schmidt, and least squares are the standard second midterm or final unit. A study sheet that starts with the characteristic polynomial before the class has defined it will crowd out row reduction, which is what this exam actually times. Markov chains and applications, if your course does them, are usually last.

The rules

What a note sheet is usually allowed to include

Linear algebra policies vary more than calculus. Some courses allow a handwritten card of theorems and ban worked examples. Some allow no notes and provide no formula sheet, because the procedures are short. A few allow a calculator only for arithmetic and not for rref. Computer-algebra row reduction is a common explicit ban even when a four-function calculator is fine. If a card is allowed, spend it on the invertible-matrix checklist and the determinant sign rules. If it is not, the study sheet is how you rehearse those checklists the day before.

The sheet

The matching library sheet

A one-page reference for the whole topic, fact-checked, with the first part free to read. This exam page does not repeat it. Use it to see the material, then make a shorter study sheet from the notes your section actually used.

Linear Algebra reference sheet — the free part in text.
Unlock the full reference sheet to edit, print or save it as a PDF.

Make your own for this exam

Upload the elimination and vector-space lectures, plus one reduced matrix you trust, so the sheet uses your textbook’s notation for free variables. Say “eigenvalues not yet” in the focus box if that chapter is in the same file. One page holds the checklists; it will not hold a 4 by 4 reduction, so keep one fully worked example in the margin only if you still have room.

Make this from your notes

FAQ

Questions

Are eigenvalues on the linear algebra midterm?: Usually they are the next exam. This midterm is row reduction, inverses, determinants, span, basis, and rank-nullity. Check whether the characteristic polynomial has been assigned.
Can I use a calculator to row-reduce?: Often no, beyond basic arithmetic. Many syllabi ban rref on a calculator or computer. Practice the hand steps even if a device is allowed for numbers.
What is the one theorem worth writing out?: The slice of the invertible matrix theorem your class proved: pivots in every row and column, only the trivial solution to Ax = 0, determinant not zero, and the columns forming a basis. Use their wording.

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