{
  "deck_name": "Linear Algebra Essentials",
  "cards": [
    { "question": "What is a vector in an introductory linear algebra course?", "answer": "An ordered list of scalars that can represent quantities such as coordinates or coefficients." },
    { "question": "What is a linear combination of vectors?", "answer": "A sum of vectors, each multiplied by a scalar coefficient." },
    { "question": "What is the dot product of two same-length vectors?", "answer": "The sum of products of corresponding components; it returns a scalar." },
    { "question": "What does the Euclidean norm of a vector measure?", "answer": "Its length, computed as the square root of the sum of squared components." },
    { "question": "What are the dimensions of an m-by-n matrix?", "answer": "It has m rows and n columns." },
    { "question": "When can an m-by-n matrix multiply an n-by-p matrix?", "answer": "When the number of columns in the first matrix equals the number of rows in the second; the result is m-by-p." },
    { "question": "What is the identity matrix's defining property?", "answer": "Multiplying a compatible vector or matrix by it leaves that object unchanged." },
    { "question": "What is the transpose of a matrix?", "answer": "The matrix formed by exchanging its rows and columns." },
    { "question": "How can a matrix-vector product be interpreted?", "answer": "As a linear combination of the matrix's columns, weighted by the vector's entries." },
    { "question": "How is a linear system commonly written in matrix form?", "answer": "A x = b, where A contains coefficients, x the variables, and b the constants." },
    { "question": "What is an augmented matrix for a linear system?", "answer": "The coefficient matrix with the right-hand-side constants appended as an extra column." },
    { "question": "Why do elementary row operations help solve a linear system?", "answer": "They produce an equivalent system with the same solution set in a form easier to analyze." },
    { "question": "What is a pivot in row-echelon form?", "answer": "The first nonzero entry in a nonzero row; pivot positions identify leading variables." },
    { "question": "What makes a linear system inconsistent?", "answer": "It has no solution, such as when row reduction yields a row representing 0 = a nonzero value." },
    { "question": "What does it mean for a set of vectors to be linearly independent?", "answer": "Only the trivial combination, with every coefficient zero, produces the zero vector." },
    { "question": "What does the span of a set of vectors contain?", "answer": "Every vector that can be expressed as a linear combination of the set." },
    { "question": "What is a basis for a vector space?", "answer": "A linearly independent set of vectors that spans the entire space." },
    { "question": "What does the dimension of a finite-dimensional vector space count?", "answer": "The number of vectors in any basis for that space." },
    { "question": "When is a square matrix invertible?", "answer": "Exactly when its associated linear transformation has an inverse; equivalently, its columns are linearly independent." },
    { "question": "What does a zero determinant tell you about a square matrix?", "answer": "The matrix is singular and does not have an inverse." }
  ]
}