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Mathematics 60 flashcards ~30 min

Linear Algebra Fundamentals

Linear algebra underlies everything from computer graphics to machine learning, and mastering its core vocabulary is the first step to fluency. This deck walks through matrices and vectors, key operations like addition, multiplication, and transposition, and i...

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Linear algebra underlies everything from computer graphics to machine learning, and mastering its core vocabulary is the first step to fluency. This deck walks through matrices and vectors, key operations like addition, multiplication, and transposition, and important properties such as determinants, rank, and invertibility. You'll also review vector spaces, linear independence, bases, and the concept of eigenvalues and eigenvectors that appear throughout advanced math and engineering. Clear, exam-style definitions make this deck a solid refresher for college linear algebra courses or a quick primer before tackling data science and applied math topics.

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A rectangular array of numbers arranged in rows and columns.
A rectangular array of numbers arranged in rows and columns.
A quantity with both magnitude and direction, often represented as an ordered list of numbers.
A quantity with both magnitude and direction, often represented as an ordered list of numbers.
The number of rows by the number of columns.
The number of rows by the number of columns.
A matrix with the same number of rows and columns.
A matrix with the same number of rows and columns.
A square matrix with 1s on the diagonal and 0s elsewhere.
A square matrix with 1s on the diagonal and 0s elsewhere.
A matrix in which all elements are zero.
A matrix in which all elements are zero.
Adding corresponding elements of two matrices of the same dimension.
Adding corresponding elements of two matrices of the same dimension.
No, the number of columns in the first must equal rows in the second.
No, the number of columns in the first must equal rows in the second.
A matrix formed by swapping its rows and columns.
A matrix formed by swapping its rows and columns.
A scalar value that indicates certain properties, like invertibility.
A scalar value that indicates certain properties, like invertibility.
A matrix with a nonzero determinant that has an inverse.
A matrix with a nonzero determinant that has an inverse.
A matrix that when multiplied by the original gives the identity matrix.
A matrix that when multiplied by the original gives the identity matrix.
A set of equations with the same variables solved simultaneously.
A set of equations with the same variables solved simultaneously.
Gaussian elimination.
Gaussian elimination.
A sum of vectors multiplied by scalar coefficients.
A sum of vectors multiplied by scalar coefficients.
A set of vectors where no vector can be written as a combination of the others.
A set of vectors where no vector can be written as a combination of the others.
A linearly independent set of vectors that spans the space.
A linearly independent set of vectors that spans the space.
The number of linearly independent rows or columns.
The number of linearly independent rows or columns.
A scalar such that Av = λv for a matrix A and vector v.
A scalar such that Av = λv for a matrix A and vector v.
A nonzero vector that only scales when a linear transformation is applied.
A nonzero vector that only scales when a linear transformation is applied.
A single numerical value, as opposed to a vector or matrix.
A single numerical value, as opposed to a vector or matrix.
The sum of the products of corresponding components.
The sum of the products of corresponding components.
Finding a vector perpendicular to two given vectors in 3D.
Finding a vector perpendicular to two given vectors in 3D.
A matrix with nonzero entries only on the main diagonal.
A matrix with nonzero entries only on the main diagonal.
A matrix equal to its own transpose.
A matrix equal to its own transpose.
It has a determinant of zero and no inverse.
It has a determinant of zero and no inverse.
A set of vectors closed under addition and scalar multiplication.
A set of vectors closed under addition and scalar multiplication.
All possible linear combinations of those vectors.
All possible linear combinations of those vectors.
A function between vector spaces that preserves addition and scalar multiplication.
A function between vector spaces that preserves addition and scalar multiplication.
The sum of the elements on its main diagonal.
The sum of the elements on its main diagonal.