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Row reduction
Gaussian elimination, rank and solving Ax = b.
Three row operations (swap, scale, add a multiple of one row to another) bring any matrix to reduced row-echelon form. The number of pivots is the rank; the reduced form of an augmented matrix reads out the solution of a linear system.
କାର୍ଯ୍ୟକାରୀ ଉଦାହରଣ · rref [[1,2,3],[4,5,6],[7,8,9]]
Rref [[1,2,3],[4,5,6],[7,8,9]]
ପଦକ୍ଷେପ କ୍ରମେ
- \left[\begin{matrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{matrix}\right]
Gauss-Jordan elimination.
- \left[\begin{matrix}1 & 2 & 3\\0 & -3 & -6\\7 & 8 & 9\end{matrix}\right]
R2 ← R2 − (4)·R1 to clear column 1.
- \left[\begin{matrix}1 & 2 & 3\\0 & -3 & -6\\0 & -6 & -12\end{matrix}\right]
R3 ← R3 − (7)·R1 to clear column 1.
- \left[\begin{matrix}1 & 2 & 3\\0 & 1 & 2\\0 & -6 & -12\end{matrix}\right]
R2 ← R2 / -3 to make the pivot 1.
- \left[\begin{matrix}1 & 0 & -1\\0 & 1 & 2\\0 & -6 & -12\end{matrix}\right]
R1 ← R1 − (2)·R2 to clear column 2.
- \left[\begin{matrix}1 & 0 & -1\\0 & 1 & 2\\0 & 0 & 0\end{matrix}\right]
R3 ← R3 − (-6)·R2 to clear column 2.
- \left[\begin{matrix}1 & 0 & -1\\0 & 1 & 2\\0 & 0 & 0\end{matrix}\right]
Reduced row-echelon form.
ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ
ଏବେ ଆପଣ ଗୋଟିଏ ସମସ୍ୟା ବାଛନ୍ତୁ, କିମ୍ବା ଆପଣଙ୍କର ନିଜ ସମସ୍ୟାକୁ ଟାଇପ କରନ୍ତୁ କିମ୍ବା ଆଙ୍କନ୍ତୁ। ପ୍ରତ୍ୟେକ ପର୍ଯ୍ୟାୟରେ, ଗୋଟିଏ ଚିତ୍ର, ଉତ୍ତର ଲୁକ୍କାୟିତ ହୋଇଥାଏ ଯେ ପର୍ଯ୍ୟନ୍ତ ଆପଣ ପଚାରନ୍ତି ନାହିଁ।
ଗୋଟିଏ ମୁକ୍ତ ଖାତା ପ୍ରତ୍ୟେକ ପାଠରେ ଟିପ୍ପଣୀ ଯୋଗ କରିଥାଏ, ଆପଣ ଯାହା ସମାପ୍ତ କରିଛନ୍ତି ତାହାର ଗୋଟିଏ ରେକର୍ଡ, ଆପଣଙ୍କର ସମାଧାନ ହୋଇଥିବା ସମସ୍ୟାଗୁଡ଼ିକ ଗୋଟିଏ ସ୍ଥାନରେ, ଏବଂ ଜଣେ ଶିକ୍ଷକ ଆପଣ ଏହି ପୃଷ୍ଠା ବିଷୟରେ ପଚାରିପାରିବେ। ଗଣିତ ନିଜେ ସମସ୍ତଙ୍କୁ ଖୋଲା ଅଛି, ଲଗଇନ ହୋଇଥିବା କିମ୍ବା ନଥିବା।
ସଦସ୍ୟତା ନିଅନ୍ତୁ ଲଗଇନଏଠାରେ ବ୍ୟବହୃତ ପ୍ରତୀକଗୁଡିକ
ସମ୍ପୂର୍ଣ୍ଣ ବର୍ଣ୍ଣନା ପାଇଁ ଯେକୌଣସି ପ୍ରତୀକକୁ ଟ୍ୟାପ କରନ୍ତୁ, ଗୋଟିଏ ଚିତ୍ର, ଏବଂ ଏଥିରେ ଥିବା ପ୍ରତ୍ୟେକ ଅକ୍ଷରର ଅର୍ଥ କଣ।
କିପରି: Row reduction
- Gauss-Jordan elimination.
- R2 ← R2 − (4)·R1 to clear column 1.
- R3 ← R3 − (7)·R1 to clear column 1.
- R2 ← R2 / -3 to make the pivot 1.
- R1 ← R1 − (2)·R2 to clear column 2.
- R3 ← R3 − (-6)·R2 to clear column 2.
- Reduced row-echelon form.
ଲୋକମାନେ ପଚାରୁଥିବା ପ୍ରଶ୍ନ
What does a determinant mean geometrically?
It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.
What is an eigenvector?
A direction the matrix does not turn: it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.
Why is matrix multiplication not commutative?
Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.
ଅଧିକ Linear Algebra
DeterminantsMatrix inverseEigenvalues and eigenvectorsMatrix multiplicationVector spaces, span and linear independenceOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis